SHSAT: Systems of linear equations
Solve systems by substitution or elimination, tell how many solutions a system has, and answer the exact quantity a question asks for.
When two equations share two unknowns, how do you find the values that make both true?
Short answer
Combine the equations so that one variable disappears, solve for the other, then substitute back. Check the pair in both equations before answering.
Know first: Solving one-variable equations, Slope-intercept form
Step 1 of 5
Two equations, two unknowns, one point that fits both
A system is two equations that must both be true. Its solution is the pair of values, or the point on a graph, that satisfies both at once.
There are two main methods. Substitution: solve one equation for a variable and put that expression into the other. Elimination: multiply one or both equations so a variable has opposite or equal coefficients, then add or subtract to remove it.
| What you see | Number of solutions | Example |
|---|---|---|
| different slopes | exactly one | y = 3x + 2 and y = −x + 6 meet at (1, 5) |
| same slope, different intercepts | none: parallel lines | y = 2x + 1 and y = 2x − 4 |
| one equation is a multiple of the other | infinitely many: the same line | x + 2y = 5 and 3x + 6y = 15 |
Grade 9 SHSAT forms cover math through Grade 8, which includes systems of linear equations.
Step 2 of 5
Eliminate one variable
Worked example: Notebooks and pens
4 notebooks and 2 pens cost $22. 3 notebooks and 4 pens cost $24. What is the price of one pen?
- A$3
- B$4
- C$6
- D$7
- 1
Write the system
4n + 2p = 22 and 3n + 4p = 24.
- 2
Match a coefficient
Double the first equation: 8n + 4p = 44.
- 3
Subtract
(8n + 4p) − (3n + 4p) = 44 − 24, so 5n = 20 and n = 4.
- 4
Find the pen price
4(4) + 2p = 22, so 2p = 6 and p = 3. Check the second purchase: 12 + 12 = 24.
- 5
See why the other choices are there
$4 is the notebook price. $6 is 2p, the cost of two pens. $7 is a notebook plus a pen.
Answer
$3
Find the wrong step
Solve x + 2y = 9 and 3x − y = 6.
- 1
From the first equation, x = 9 − 2y.
- 2
Substitute: 3(9 − 2y) − y = 6 becomes 27 − 2y − y = 6.
What went wrong
The 3 multiplies both terms in the parentheses, so 3(9 − 2y) = 27 − 6y.
- 3
Then −3y = −21, so y = 7 and x = 9 − 14 = −5.
The fix
27 − 6y − y = 6 gives −7y = −21, so y = 3 and x = 9 − 6 = 3. Check: 3 + 6 = 9 and 9 − 3 = 6.
Step 3 of 5
Solve, then answer what is asked
Check yourself · Question 1
The lines y = 5x − 4 and y = −x + 14 intersect. What is the y-coordinate of the point where they meet?
Answer: B
Set the outputs equal: 5x − 4 = −x + 14, so 6x = 18 and x = 3. Then y = 5(3) − 4 = 11. Check: −3 + 14 = 11.
Trap answer: 3
Reporting the wrong coordinate
- Why it looks right
- Solving for x is the main work, so x feels like the answer.
- Why it's wrong
- The question asks for the y-coordinate. Substitute x = 3 into either equation.
- The right answer
- 11, from 5(3) − 4.
Check yourself · Question 2
For what value of k does the system 3x − 2y = 7 and kx − 4y = 5 have no solution?
Answer: C
No solution means parallel lines: the x and y coefficients must be in the same ratio but the constants must not. −4 is 2 times −2, so k = 2 × 3 = 6. Then the second equation is 6x − 4y = 5, while doubling the first gives 6x − 4y = 14.
Check yourself · Question 3
If x + y = 13 and 3x − y = 11, what is the value of x − y?
Answer: A
Add the equations to eliminate y: 4x = 24, so x = 6. Then y = 13 − 6 = 7, and x − y = 6 − 7 = −1.
Common mistake
I stop as soon as I find one variable.
- Why it's tempting
- Finding the first value takes most of the work, and it often appears among the choices.
- Do this instead
- Reread the question before choosing. It may ask for the other variable, a combination such as x − y, or a total cost.
Step 4 of 5
What to remember
Remember
- Use substitution when a variable is easy to isolate, and elimination when coefficients can be matched.
- Different slopes give one solution, parallel lines give none, and the same line gives infinitely many.
- Check the pair in both equations, then report exactly what the question asks for.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, which variable did you eliminate or substitute first, and how did you check your answer in both equations?